This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and geometry of complex numbers, and then we will make our way via differentiation, integration, complex dynamics, power series representation and Laurent series into territories at the edge of what is known today. Each module consists of five video lectures with embedded quizzes, followed by an electronically graded homework assignment. Additionally, modules 1, 3, and 5 also contain a peer assessment.
The homework assignments will require time to think through and practice the concepts discussed in the lectures. In fact, a significant amount of your learning will happen while completing the homework assignments. These assignments are not meant to be completed quickly; rather you'll need paper and pen with you to work through the questions. In total, we expect that the course will take 6-12 hours of work per module, depending on your background.

From the lesson

Laurent Series and the Residue Theorem

Laurent series are a powerful tool to understand analytic functions near their singularities. Whereas power series with non-negative exponents can be used to represent analytic functions in disks, Laurent series (which can have negative exponents) serve a similar purpose in annuli. We’ll begin this module by introducing Laurent series and their relation to analytic functions and then continue on to the study and classification of isolated singularities of analytic functions. We’ll encounter some powerful and famous theorems such as the Theorem of Casorati-Weierstraß and Picard’s Theorem, both of which serve to better understand the behavior of an analytic function near an essential singularity. Finally we’ll be ready to tackle the Residue Theorem, which has many important applications. We’ll learn how to find residues and evaluate some integrals (even some real integrals on the real line!) via this important theorem.